Control problem for diffusion-type random fields
The following optimal control problem for a stochastic differential equation \[ \xi (t,s) = \xi_0 + \int^t_0 \int^s_0 a(x,y, \xi, u) dx dy + \int^t_0 \int^s_0 b(x,y, \xi) W(dx dy), \] \[ F(u) = \mathbb{E} \Biggl( \int^1_0 \int^1_0 f(t,s, \xi^u (t,s), u(t,s) \xi^u (t,s)) dt ds \Biggr) \to \min \] \vskip2mm is considered. For this problem, sufficient conditions for the existence of optimal controls, derived by Girsanov's method of transformation of measures, are given. Also the existence of \(\varepsilon\)-optimal controls is proved and a method for their construction is described.
scientific article; zbMATH DE number 844530